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Properties

Label 880.2.bk.b
Level $880$
Weight $2$
Character orbit 880.bk
Analytic conductor $7.027$
Analytic rank $0$
Dimension $236$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(243,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.243");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bk (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(118\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 236 q + 4 q^{3} + 4 q^{5} - 4 q^{7} + 12 q^{8} + 240 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 236 q + 4 q^{3} + 4 q^{5} - 4 q^{7} + 12 q^{8} + 240 q^{9} - 8 q^{12} + 12 q^{14} - 4 q^{15} - 8 q^{16} + 8 q^{17} + 28 q^{18} + 28 q^{19} + 4 q^{20} + 16 q^{21} + 4 q^{22} - 8 q^{23} + 12 q^{25} + 8 q^{26} + 4 q^{27} - 36 q^{28} + 8 q^{29} - 56 q^{30} - 40 q^{32} - 4 q^{33} - 44 q^{34} - 28 q^{35} - 32 q^{36} - 12 q^{38} - 40 q^{40} - 20 q^{42} - 40 q^{46} - 48 q^{47} - 40 q^{48} - 28 q^{51} - 4 q^{52} + 28 q^{53} - 24 q^{56} - 16 q^{57} - 76 q^{58} + 8 q^{59} - 4 q^{60} - 24 q^{61} + 12 q^{62} - 24 q^{63} - 32 q^{65} - 4 q^{66} + 36 q^{68} + 48 q^{69} + 28 q^{70} - 4 q^{71} + 20 q^{72} + 8 q^{73} - 68 q^{75} - 24 q^{76} + 12 q^{77} + 8 q^{78} + 40 q^{79} + 112 q^{80} + 244 q^{81} - 64 q^{82} + 24 q^{83} - 104 q^{84} + 8 q^{85} + 72 q^{86} - 140 q^{87} - 32 q^{88} + 36 q^{89} - 136 q^{90} + 12 q^{91} + 72 q^{92} - 16 q^{94} - 36 q^{95} + 72 q^{96} + 20 q^{97} + 136 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
243.1 −1.41227 − 0.0741213i −1.20793 1.98901 + 0.209359i −1.52213 − 1.63802i 1.70593 + 0.0895337i 1.78958 − 1.78958i −2.79350 − 0.443099i −1.54090 2.02825 + 2.42615i
243.2 −1.41215 − 0.0764611i 1.79727 1.98831 + 0.215948i 1.04979 + 1.97432i −2.53801 − 0.137422i −3.51807 + 3.51807i −2.79127 − 0.456979i 0.230194 −1.33150 − 2.86829i
243.3 −1.41059 + 0.101209i −1.19235 1.97951 − 0.285529i −0.235780 − 2.22360i 1.68191 − 0.120677i −2.51302 + 2.51302i −2.76338 + 0.603108i −1.57830 0.557637 + 3.11272i
243.4 −1.40602 − 0.152014i 1.42312 1.95378 + 0.427470i 0.506935 − 2.17785i −2.00093 − 0.216334i 2.96441 − 2.96441i −2.68208 − 0.898034i −0.974737 −1.04382 + 2.98503i
243.5 −1.40275 − 0.179717i 0.795514 1.93540 + 0.504194i −1.84699 + 1.26041i −1.11591 − 0.142967i −0.705976 + 0.705976i −2.62427 − 1.05508i −2.36716 2.81738 − 1.43610i
243.6 −1.39926 + 0.205127i 0.517505 1.91585 − 0.574051i 1.82639 + 1.29008i −0.724123 + 0.106154i 1.67327 − 1.67327i −2.56301 + 1.19624i −2.73219 −2.82022 − 1.43052i
243.7 −1.39674 + 0.221632i −1.99322 1.90176 − 0.619123i 1.27472 + 1.83714i 2.78400 − 0.441760i 0.168443 − 0.168443i −2.51904 + 1.28624i 0.972916 −2.18762 − 2.28349i
243.8 −1.39301 + 0.243961i 1.29110 1.88097 − 0.679680i −1.85420 + 1.24978i −1.79852 + 0.314979i 1.21062 − 1.21062i −2.45439 + 1.40568i −1.33305 2.27802 − 2.19331i
243.9 −1.38403 − 0.290618i −1.48289 1.83108 + 0.804450i 2.23341 − 0.109047i 2.05237 + 0.430957i −2.19311 + 2.19311i −2.30049 − 1.64553i −0.801024 −3.12280 − 0.498145i
243.10 −1.38402 + 0.290657i −3.12744 1.83104 − 0.804551i −2.12238 + 0.703930i 4.32844 − 0.909011i −1.29734 + 1.29734i −2.30035 + 1.64572i 6.78086 2.73281 − 1.59114i
243.11 −1.35246 − 0.413341i 2.75998 1.65830 + 1.11806i 2.15400 − 0.600239i −3.73277 − 1.14082i −0.140016 + 0.140016i −1.78064 − 2.19757i 4.61751 −3.16130 − 0.0785374i
243.12 −1.35000 + 0.421294i 3.23238 1.64502 − 1.13750i −0.0918916 + 2.23418i −4.36373 + 1.36178i 1.78612 − 1.78612i −1.74157 + 2.22866i 7.44831 −0.817191 − 3.05486i
243.13 −1.34156 − 0.447453i −3.25522 1.59957 + 1.20057i 0.702554 − 2.12283i 4.36708 + 1.45656i −0.410588 + 0.410588i −1.60872 − 2.32637i 7.59648 −1.89239 + 2.53355i
243.14 −1.33327 − 0.471597i −2.25068 1.55519 + 1.25753i −1.07760 + 1.95928i 3.00075 + 1.06141i −1.22152 + 1.22152i −1.48044 − 2.41004i 2.06557 2.36072 − 2.10404i
243.15 −1.28261 + 0.595739i −0.00646194 1.29019 − 1.52820i 1.81833 − 1.30141i 0.00828816 − 0.00384963i −1.60179 + 1.60179i −0.744402 + 2.72871i −2.99996 −1.55692 + 2.75245i
243.16 −1.28139 − 0.598363i −1.03090 1.28392 + 1.53347i −0.753550 + 2.10527i 1.32099 + 0.616852i −0.0342787 + 0.0342787i −0.727636 − 2.73323i −1.93724 2.22531 − 2.24678i
243.17 −1.27343 − 0.615122i 2.74354 1.24325 + 1.56663i −2.17093 − 0.535800i −3.49371 − 1.68761i −0.431323 + 0.431323i −0.619521 − 2.75975i 4.52702 2.43494 + 2.01769i
243.18 −1.23734 + 0.684833i 0.322247 1.06201 − 1.69474i −2.23514 + 0.0642867i −0.398728 + 0.220685i −3.32902 + 3.32902i −0.153447 + 2.82426i −2.89616 2.72160 − 1.61024i
243.19 −1.23570 − 0.687779i 0.696721 1.05392 + 1.69978i 0.908836 − 2.04304i −0.860940 − 0.479190i −1.05162 + 1.05162i −0.133260 − 2.82529i −2.51458 −2.52821 + 1.89951i
243.20 −1.23415 + 0.690563i −1.48151 1.04624 − 1.70452i −1.55039 + 1.61130i 1.82841 − 1.02308i 2.89237 − 2.89237i −0.114146 + 2.82612i −0.805118 0.800715 − 3.05922i
See next 80 embeddings (of 236 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 243.118
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bk.b yes 236
5.c odd 4 1 880.2.s.b ✓ 236
16.f odd 4 1 880.2.s.b ✓ 236
80.s even 4 1 inner 880.2.bk.b yes 236
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.s.b ✓ 236 5.c odd 4 1
880.2.s.b ✓ 236 16.f odd 4 1
880.2.bk.b yes 236 1.a even 1 1 trivial
880.2.bk.b yes 236 80.s even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{118} - 2 T_{3}^{117} - 235 T_{3}^{116} + 468 T_{3}^{115} + 26789 T_{3}^{114} + \cdots + 441442156675072 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display